This is a master course given in Moscow at the Laboratory of Algebraic Geometry of the National Research University Higher School of Economics by Valery Gritsenko, a professor of University Lille 1, France.
Jacobi forms are holomorphic functions in two complex variables. They are modular in one variable and abelian (or double periodic) in another variable. The theory of Jacobi modular forms became an independent research subject after the famous book of Martin Eichler and Don Zagier “Jacobi modular forms” (Progress in Mathematics, vol. 55, 1985) which was cited more than a thousand times in research papers. This is due to many applications of Jacobi forms in arithmetic, topology, algebraic and differential geometry, mathematical and theoretical physics, in the theory of Lie algebras, etc. The list of mentioned subjects shows that my course might be useful for master and Ph.D. students working in different directions.
Motivated undergraduate students can also study this subject. To follow the course one has to know only elementary basic facts from the theory of modular forms (for example, the paragraphs 1-4 of the chapter VII of Serre’s “A Course in Arithmetic” are enough).
The main hero of the course is the Jacobi theta-series. Using it we will construct a lot of concrete examples of Jacobi forms in one or many abelian variables, in particular, Jacobi forms for root systems.
For some of you, who will be successful with the theoretical exercises of the course, I am ready to formulate research problems for Master or Ph.D. thesis. (Ph.D. support might be available at CEMPI in Lille or at the Faculty of Mathematics of National Research University Higher School of Economics in Moscow)

100 % en línea

100 % en línea

Comienza de inmediato y aprende a tu propio ritmo.

Fechas límite flexibles

Fechas límite flexibles

Restablece las fechas límite en función de tus horarios.

Nivel avanzado

Nivel avanzado

Horas para completar

Aprox. 54 horas para completar

Sugerido: 12 weeks of study, 3-5 hours per week...

Idiomas disponibles

Inglés (English)

Subtítulos: Inglés (English)

100 % en línea

100 % en línea

Comienza de inmediato y aprende a tu propio ritmo.

Fechas límite flexibles

Fechas límite flexibles

Restablece las fechas límite en función de tus horarios.

Nivel avanzado

Nivel avanzado

Horas para completar

Aprox. 54 horas para completar

Sugerido: 12 weeks of study, 3-5 hours per week...

Idiomas disponibles

Inglés (English)

Subtítulos: Inglés (English)

Programa - Qué aprenderás en este curso

Semana

1

Horas para completar

1 hora para completar

Introduction to the Course

Welcome to the course! I hope you have an opportunity to reserve some time to explore the course content, course logic and our grading policy. The course consists of 12 lectures. This course will help you to start your progress in the field of the theory of Jacobi modular forms.
Best regards, Valery Gritsenko...

Zeros of Jacobi forms. The Jacobi theta-series, the Dedekind eta-function and the first examples of Jacobi modular forms

This module is devoted to zeros of Jacobi modular forms, their Taylor extensions and the first examples of Jacobi forms. Using classical Jacobi theta-series and Dedekind eta-function we construct a series of Jacobi forms. Also there is a peer review in the end of this module....

Instructor

Acerca de National Research University Higher School of Economics

National Research University - Higher School of Economics (HSE) is one of the top research universities in Russia. Established in 1992 to promote new research and teaching in economics and related disciplines, it now offers programs at all levels of university education across an extraordinary range of fields of study including business, sociology, cultural studies, philosophy, political science, international relations, law, Asian studies, media and communications, IT, mathematics, engineering, and more.
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